Dynamics of a periodic bluetongue model with a temperature-dependent incubation period. (English) Zbl 1437.34080
Summary: Bluetongue is a midge-borne disease that is transmitted by biting midges of the Culicoides family. In this paper, we propose a bluetongue model with seasonality and temperature-dependent incubation period. We introduce the basic disease reproduction ratio for the whole system \(R_0\) and the basic disease reproduction ratio in the absence of sheep \(\widetilde{R}_0\), and establish the threshold type results on the global dynamics in terms of \(R_0\) and \(\widetilde{R}_0\). More precisely, bluetongue persists in cattle and midges but it may eradicate the sheep if \(R_0>1\) and \(\widetilde{R}_0 >1\); the system admits a positive periodic solution, the disease is uniformly persistent in cattle, sheep, and midges, and bluetongue cannot eradicate the sheep if \(R_0>1\geq \widetilde{R}_0\). Numerically, we study the bluetongue transmission case in France, and perform some sensitivity analysis of \(R_0\). Our numerical simulations are carried out to illustrate the analytic results.
MSC:
| 34K60 | Qualitative investigation and simulation of models involving functional-differential equations |
| 92D30 | Epidemiology |
| 34K13 | Periodic solutions to functional-differential equations |
| 34K25 | Asymptotic theory of functional-differential equations |
Keywords:
midge-borne disease; periodic incubation period; basic reproduction ratio; periodic solution; uniform persistenceReferences:
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